cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A370993 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x*log(1-x)) ).

Original entry on oeis.org

1, 0, 2, 3, 80, 450, 11424, 133140, 3670400, 68303088, 2123212320, 54742984560, 1938915574848, 63653459126400, 2565847637273088, 101718189575664480, 4637150408792355840, 214393171673968519680, 10962579011721928980480, 577166004742408670937600
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052802, A370994, A370995.
Cf. A052830, A370988.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x*log(1-x)))/x))
  • PARI
    a(n) = sum(k=0, n\2, (n+k)!*abs(stirling(n-k, k, 1))/(n-k)!)/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (n+k)! * |Stirling1(n-k,k)|/(n-k)!.

A371119 E.g.f. satisfies A(x) = 1 + x*A(x)*(exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 2, 3, 52, 305, 4866, 57337, 1048776, 18547713, 407900710, 9436057961, 248501026236, 7021087254337, 217488458525898, 7223642070331065, 258233053457437456, 9841074705853124609, 399304906991091898830, 17163110041947804495817, 779646387683354742170820
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Links

Crossrefs

Cf. A370988, A371115, A371120.
Cf. A000272, A371139, A376293.
Cf. A371121.

Programs

  • PARI
    a(n) = n!^2*sum(k=0, n\2, stirling(n-k, k, 2)/((n-k)!*(n-k+1)!));

Formula

a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} Stirling2(n-k,k)/( (n-k)! * (n-k+1)! ).
E.g.f.: (1/x) * Series_Reversion( x/(1 + x*(exp(x) - 1)) ). - Seiichi Manyama, Sep 19 2024

A370989 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2*(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 2910, 22722, 117656, 8482392, 143398170, 1519998590, 79655138772, 2206506673956, 39101112995126, 1798446230741370, 68667380639283120, 1795441154500375472, 81344029377887798706, 3830461514154681289974, 135388937631209203030700
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052894, A370988, A370990.
Cf. A358013.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2*(exp(x)-1)))/x))
  • PARI
    a(n) = sum(k=0, n\3, (n+k)!*stirling(n-2*k, k, 2)/(n-2*k)!)/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n+k)! * Stirling2(n-2*k,k)/(n-2*k)!.

A376345 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*(exp(x^2) - 1)) ).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 2880, 840, 201600, 7998480, 12700800, 1816547040, 67898476800, 311359688640, 35628798965760, 1317155266627200, 12924530383564800, 1308998905659244800, 49463008450023168000, 863080350836537433600, 81264621182097120768000, 3227330594664084337228800, 87828327888763088096870400
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2024

Keywords

Links

Crossrefs

Cf. A370988, A376347.
Cf. A370989, A375588.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*(exp(x^2)-1)))/x))
  • PARI
    a(n) = sum(k=0, n\2, (2*n-2*k)!*stirling(k, n-2*k, 2)/k!)/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (2*n-2*k)! * Stirling2(k,n-2*k)/k!.

A370990 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^3*(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 201936, 1996344, 12701520, 64865790, 17053788840, 374788816116, 4944496679304, 50034166184730, 6390396135006240, 239770550508132720, 5363062998193560096, 89908444484550625014, 7402557588108228698040
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052894, A370988, A370989.
Cf. A358014.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^3*(exp(x)-1)))/x))
  • PARI
    a(n) = sum(k=0, n\4, (n+k)!*stirling(n-3*k, k, 2)/(n-3*k)!)/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (n+k)! * Stirling2(n-3*k,k)/(n-3*k)!.

A370991 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2/2*(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 735, 5691, 29428, 1122696, 18159165, 190810675, 5768268726, 143497346928, 2479363382587, 73013461310895, 2336253676913640, 58015822633914736, 1850758447642034553, 69357415099500398979, 2252468410247071488970
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A370988, A370992.
Cf. A353998.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2/2*(exp(x)-1)))/x))
  • PARI
    a(n) = sum(k=0, n\3, (n+k)!*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n+k)! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).

A370992 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^3/6*(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 5656, 55524, 352920, 1801965, 85636540, 1762160686, 22992890284, 232001269955, 6581012518640, 197506018950920, 4224661065644016, 69931313468126169, 1757395269147356340, 60785516594782517650, 1818493252905482003620
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A370988, A370991.
Cf. A353999.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^3/6*(exp(x)-1)))/x))
  • PARI
    a(n) = sum(k=0, n\4, (n+k)!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (n+k)! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A371120 E.g.f. satisfies A(x) = 1 + x*A(x)^3*(exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 2, 3, 100, 545, 17946, 203497, 7194440, 132963777, 5172409630, 135827977241, 5868623306844, 200952952956769, 9665278822378466, 407661518051710665, 21789972653746494736, 1088515671895571005313, 64406426353877958253254
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Cf. A370988, A371115, A371119.
Cf. A371122.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (n+2*k)!*stirling(n-k, k, 2)/((n-k)!*(n+k+1)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n+2*k)! * Stirling2(n-k,k)/( (n-k)! * (n+k+1)! ).

A371273 E.g.f. satisfies A(x) = 1 + x*A(x)^4 * (exp(x*A(x)^3) - 1).

Original entry on oeis.org

1, 0, 2, 3, 172, 1025, 54606, 710017, 38964024, 855167553, 49992166090, 1603665906161, 101454726848388, 4342187407054081, 299554876119595110, 16084216120063348545, 1213404824364026124016, 78279943651487041769345, 6456915976418046368634402
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2024

Keywords

Crossrefs

Cf. A370988, A371271.
Cf. A371232.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (3*n+k)!*stirling(n-k, k, 2)/(n-k)!)/(3*n+1)!;

Formula

a(n) = (n!/(3*n+1)!) * Sum_{k=0..floor(n/2)} (3*n+k)! * Stirling2(n-k,k)/(n-k)!.

A376347 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*(exp(x^3) - 1)) ).

Original entry on oeis.org

1, 0, 0, 0, 24, 0, 0, 2520, 201600, 0, 604800, 259459200, 16765056000, 259459200, 406832025600, 100037089152000, 5963169474662400, 844757641728000, 560207699251200000, 107716905363549081600, 6157546579533533184000, 3525275009847951360000, 1582967914636148232192000, 264668100119565849907200000
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2024

Keywords

Links

Crossrefs

Cf. A370988, A376345.
Cf. A370928, A375589.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*(exp(x^3)-1)))/x))
  • PARI
    a(n) = sum(k=0, n\3, (2*n-3*k)!*stirling(k, n-3*k, 2)/k!)/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (2*n-3*k)! * Stirling2(k,n-3*k)/k!.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.